45.12. Let n = 589. Note that n is the product of two primes. Most values in Z, have nine distinct cube roots. In this exercise we ask you to develop a way to factor n if, somehow, you were given all the cube roots of some a e Zn. Indeed, 201 is a perfect cube in Zm and here are all of its cube roots: 17, 54, 271, 301, 302, 358, 518, 549, and 575. Show how you can use these values to calculate the cube root of n. Note: We chose a small integer n to make your calculations feasible to perform with a calculator. Because n is so small, it might be easier just to use trial division. However, the point of this exercise is to show how it is possible to factor n quickly when given all nine cube roots of a perfect cube.
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Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
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